Tools and Techniques For Analysis of Nonconforming Elements

I thought these results would make for a great blog post. I hope this can help someone out there!

Inverse Inequalities

We reproduce the inequalities for convenience:

These rely mainly on trace-type results and scaling arguments.

First Inequality Proof

For the first we have:

by the trace theorem and a scaling argument. Rearranging:

Now since is proportional to the area of , then:

Noticing that completes the result.

Second Inequality Proof

For the second:

Now since by the formula for the area of a triangle, multiplying by on both sides yields:

Error analysis for nonconforming finite elements

We demonstrate how to obtain a linear convergence rate for nonconforming finite elements for a simple Poisson problem in 2 dimensions where the source term is given by a mesh-aligned function. Letting denote the unit square, and a conforming shape-regular triangulation thereof and its mesh skeleton. Let denote a subset of . For simplicity, we define the functional by:

where is a smooth polynomial defined on . The variational system we study is:

which corresponds to the strong form:

Using integration by parts:

The discrete problem for (nonconforming ) is:

where is the average operator. For :

Using the identity :

Applying the Cauchy-Schwarz inequality and inverse estimates:

Combining consistency and approximation errors via a Strang-type argument:

Scaling Arguments

Let be convex simplices in with an affine diffeomorphism and Jacobian :

For and :

Proof Sketch for :

For :

For :

Interpolation Error Estimate:

Using the Bramble-Hilbert theorem: